\[ \int \frac {x^4}{\coth ^{-1}(\tanh (a+b x))^3} \, dx \]
Optimal antiderivative \[ \frac {3 x^{2}}{b^{3}}+\frac {6 x \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{b^{4}}-\frac {x^{4}}{2 b \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}-\frac {2 x^{3}}{b^{2} \mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}+\frac {6 \left (b x -\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )^{2} \ln \left (\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )\right )}{b^{5}} \]
command
integrate(x^4/arccoth(tanh(b*x+a))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {16 \, \pi ^{3} b x - 96 i \, \pi ^{2} a b x - 192 \, \pi a^{2} b x + 128 i \, a^{3} b x + 7 i \, \pi ^{4} + 56 \, \pi ^{3} a - 168 i \, \pi ^{2} a^{2} - 224 \, \pi a^{3} + 112 i \, a^{4}}{-32 i \, b^{7} x^{2} + 32 \, \pi b^{6} x - 64 i \, a b^{6} x + 8 i \, \pi ^{2} b^{5} + 32 \, \pi a b^{5} - 32 i \, a^{2} b^{5}} + \frac {x^{2}}{2 \, b^{3}} - \frac {3 \, {\left (i \, \pi + 2 \, a\right )} x}{2 \, b^{4}} - \frac {3 \, {\left (\pi ^{2} - 4 i \, \pi a - 4 \, a^{2}\right )} \log \left (i \, \pi + 2 \, b x + 2 \, a\right )}{2 \, b^{5}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{4}}{\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{3}}\,{d x} \]________________________________________________________________________________________